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Section 12.13 : Spherical Coordinates

6. Convert the equation written in Spherical coordinates into an equation in Cartesian coordinates.

\[\csc \varphi = 2\cos \theta + 4\sin \theta \]

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Start Solution

There really isn’t a whole lot to do here. All we need to do is to use the following conversion formulas in the equation where (and if) possible

\[\begin{array}{c}x = \rho \sin \varphi \cos \theta \hspace{0.5in}y = \rho \sin \varphi \sin \theta \hspace{0.5in}z = \rho \cos \varphi \\ {\rho ^2} = {x^2} + {y^2} + {z^2}\end{array}\] Show Step 2

To make this problem a little easier let’s first do some rewrite on the equation.

First, let’s deal with the cosecant.

\[\frac{1}{{\sin \varphi }} = 2\cos \theta + 4\sin \theta \hspace{0.25in}\,\,\, \to \hspace{0.5in}1 = 2\sin \varphi \cos \theta + 4\sin \varphi \sin \theta \]

Next, let’s multiply everything by \(\rho \) to get,

\[\rho = 2\rho \sin \varphi \cos \theta + 4\rho \sin \varphi \sin \theta \]

Doing this makes recognizing the terms on the right a little easier.

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Using the appropriate conversion formulas from Step 1 gives,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {{x^2} + {y^2} + {z^2}} = 2x + 4y}}\]